The study of spatially periodic solutions is also important and interesting in the theory of conservation laws. For instance, the periodic solutions of hyperbolic conservation laws lose bounded total variations and are not able to be constructed through the celebrated Glimm scheme [1]}. In that respect, the genuinely nonlinear conservation laws of at most two equations are exceptional; see [2]}, [3]}, [4]}. And for the \( N\times N \)  hyperbolic systems with \( N\ge 3 \) , for instance, the non-isentropic compressible Euler equations possessing totally different nonlinear effect from the isentropic case, the global-in-time existence of periodic solutions is still open so far.
We refer to [5]}, [6]} for a large time existence and to [7]}, [8]} for the study of a resonance phenomenon of periodic solutions, respectively.
Besides the periodic solutions which oscillate around constant states, the periodic perturbations of general Riemann solutions, especially for the shock waves, are also full of interest to study.
In that case, the initial data oscillate around different constant states at the far field and therefore are no longer periodic on the whole space. For the scalar conservation laws, [9]}, [10]} and [11]} proved the nonlinear stability of shocks and rarefaction waves in both inviscid and viscous cases, based on the generalized characteristics analysis and a comparison principle, respectively.
The results reveal that different from the localized (e.g. compactly supported) perturbations in [12]}, the periodic oscillations generate a different kind of shifts to the viscous shocks, which depend on the nonlinear fluxes of conservation laws in general. Recently, the study was further continued through the energy method to some more important models such as the compressible N-S equations and multi-d scalar viscous conservation laws; see [13]}, [14]}, [15]}, [16]}, [17]}.
